Algebra

What Is the Discriminant? Reading a Quadratic at a Glance

By Math Solving Space · · 2 min read

On this page
  1. Where it comes from
  2. The three cases
  3. Worked example 1: two real roots
  4. Worked example 2: one repeated root
  5. Worked example 3: no real roots
  6. Why it matters
  7. Common mistakes
  8. Practice questions
  9. Frequently asked questions

The discriminant of ax2+bx+c=0ax^2 + bx + c = 0 is the number Δ=b2−4ac\Delta = b^2 - 4ac. If it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative there are no real roots, only complex ones. You can find this out before solving anything.

Where it comes from

The quadratic formula is

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Everything depends on the square root. The expression under it, b2−4acb^2 - 4ac, decides what kind of number the square root is.

The three cases

Δ=b2−4ac\Delta = b^2 - 4ac Roots Graph of y=ax2+bx+cy = ax^2 + bx + c
Δ>0\Delta > 0 Two different real roots Crosses the x-axis twice
Δ=0\Delta = 0 One repeated real root Touches the x-axis once
Δ<0\Delta < 0 Two complex roots Never meets the x-axis

Worked example 1: two real roots

2x2+3x−2=02x^2 + 3x - 2 = 0 has a=2a = 2, b=3b = 3, c=−2c = -2:

Δ=32−4(2)(−2)=9+16=25\Delta = 3^2 - 4(2)(-2) = 9 + 16 = 25

25>025 > 0, so there are two real roots. Better still, 2525 is a perfect square, so the roots are rational: x=−2x = -2 and x=12x = \frac{1}{2}.

Worked example 2: one repeated root

x2−6x+9=0x^2 - 6x + 9 = 0:

Δ=(−6)2−4(1)(9)=36−36=0\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0

One repeated root: x=62=3x = \frac{6}{2} = 3. Indeed x2−6x+9=(x−3)2x^2 - 6x + 9 = (x - 3)^2.

Worked example 3: no real roots

x2+2x+5=0x^2 + 2x + 5 = 0:

Δ=22−4(1)(5)=4−20=−16\Delta = 2^2 - 4(1)(5) = 4 - 20 = -16

Negative, so no real roots. Using −16=4i\sqrt{-16} = 4i:

x=−2±4i2=−1±2ix = \frac{-2 \pm 4i}{2} = -1 \pm 2i

The parabola sits entirely above the x-axis.

Why it matters

  • Exams often ask "how many solutions?" without asking you to solve. The discriminant answers that in one line.
  • Word problems: if a projectile question gives Δ<0\Delta < 0 for "when does it reach 50 m?", the object never reaches that height.
  • Tangency: a line touches a curve exactly when the resulting quadratic has Δ=0\Delta = 0.

Common mistakes

  • Forgetting to square a negative bb correctly: (−6)2=36(-6)^2 = 36, not −36-36.
  • Mixing up the sign of cc: with c=−2c = -2, −4ac-4ac becomes +16+16.
  • Using the discriminant before rearranging to =0= 0.

Practice questions

  1. How many real roots does x2+4x+4=0x^2 + 4x + 4 = 0 have?
  2. How many real roots does 3x2−x+1=03x^2 - x + 1 = 0 have?
  3. For which kk does x2+kx+9=0x^2 + kx + 9 = 0 have one repeated root?

Answers: 1) One (Δ=0\Delta = 0) 2) None (Δ=−11\Delta = -11) 3) k=6k = 6 or k=−6k = -6

Try your own quadratics in the quadratic equation solver, which shows the discriminant, and explore complex roots in the complex number calculator. For full methods, see three ways to solve a quadratic equation.

Frequently asked questions

Is the discriminant the same as the quadratic formula?

No. It is just the part under the square root, b² − 4ac. It tells you about the roots without finding them.

Can the discriminant tell me the roots are rational?

Yes. If a, b and c are whole numbers and the discriminant is a perfect square, the roots are rational.

What does Δ stand for?

Δ (delta) is the usual symbol for the discriminant. Some books write D instead.

#quadratics#algebra

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